This document is a lesson on order of operations in mathematics. It discusses the proper order for evaluating expressions with multiple operations such as addition, subtraction, multiplication, division, exponents, and parentheses. Exponents have the highest precedence and should be evaluated first, followed by multiplication and division from left to right, then addition and subtraction from left to right. Operations within parentheses are performed first, regardless of exponent operations. The document provides examples of evaluating expressions and exercises for students to practice order of operations.

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Problem Set Lesson 5 Answers

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Problem Set Lesson 5 Answers ­ Continued

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MODULE 4 Expressions and Equaons
Topic B: Special Notaons of Operaons
Lesson 6: Order of Operaon
Student Outcomes
§ Students evaluate numerical expressions. They recognize that in the absence of parentheses,
exponents are evaluated first.

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Evaluate
3 + 4 x 2
Only one of these can be correct. When we evaluate expressions, we must
agree to use one set of rules so that everyone arrives at the same correct
answer.

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§ During the last lesson, we said that addion was a shortcut to “counng on.”
How could you think about subtracon?
ú Subtracon is a shortcut to “counng back.”
§ These were the first operaons that you learned because they are the least
complicated. Next, you learned about mulplicaon and division.
§ Mulplicaon can be thought of as repeated addion. Thinking back on Lesson
4, how could you think about division?
ú Division is repeated subtracon.

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Addion and subtracon are at the same level in the order of operaons and are
evaluated from le to right in an expression. Now that these rules of Order of
Operaons are clear, can you go back and evaluate the expression 3 + 4 x 2 = 11
Mulplicaon and division are more powerful than addion and subtracon, which
led mathemacians to develop the order of operaons in this way. When we
evaluate expressions that have any of these four operaons, we always calculate
mulplicaon and division before doing any addion or subtracon. Since
mulplicaon and division are equally powerful, we simply evaluate these two
operaons as they are wrien in the expression, from le to right.

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Classwork
Example 1: Expressions with Only Addion, Subtracon, Mulplicaon, and Division
What operaons are evaluated first?
What operaons are always evaluated last?

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Exercises
1. 4 + 2 x 7
2. 36 ÷ 3 x 4
3. 20 – 5 x 2

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§ In the last lesson, you learned about exponents, which are a way of wring repeated
mulplicaon. So, exponents are more powerful than mulplicaon or division. If exponents are
present in an expression, they are evaluated before any mulplicaon or division.
§ When we evaluate expressions, we must agree to use one set of rules so that everyone arrives at
the same correct answer. These rules are based on doing the most powerful operaons first
(exponents), then the less powerful ones (mulplicaon and division, going from le to right), and
finally the least powerful ones last (addion and subtracon, going from le to right).
Evaluate the expression
4 + 6 x 6 ÷ 8
Now evaluate the expression
4 + 62 ÷ 8

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Example 2: Expressions with Four Operaons and Exponents
4 + 92 ÷ 3 x 2 ‐ 2
What operaon is evaluated first?
What operaons are evaluated next?
What operaons are always evaluated last?
What is the final answer?

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Exercises
3. 90 – 52 x 3 4. 43 + 2 x 8

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The last important rule in the order of operaons involves grouping symbols (usually
parentheses). These tell us that in certain circumstances or scenarios, we need to do
things out of the usual order. Operaons inside grouping are always evaluated first,
before exponents.
Consider a family of 4 that goes to a soccer game. Tickets are $5.00
each. The mom also buys a so drink for $2.00 How would you write
this expression?
How much will this oung cost?

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Consider a different scenario: the family goes to the game like before, but each of
the family members wants a drink. How would you write this expression?
Why would you add the 5 and 2 first?
How much will this oung cost?
How many groups are there?
What is each group comprised of?

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§ The last complicaon that can arise is that if two or more sets of parentheses
are ever needed, evaluate the innermost parentheses first, and work outward.
§ Try Exercises 6 and 7.
Exercises
6. 2 + (92 – 4) 7. 2 ∙ (13 + 5 – 14 ÷ (3 + 4))

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Let’s take a look at how parentheses and exponents work together. Somemes a
problem will have parentheses and the values inside the parentheses have an
exponent. Let’s evaluate this expression:
Example 4: Expressions with Parentheses and Exponents
2 x (3 + 42)
Which value will we evaluate first within the parentheses? Evaluate.
Evaluate the rest of the expression.

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What do you think will happen when the exponent in this expression is outside of the parentheses?
2 x (3 + 4)2
Will the answer be the same?
Which should we evaluate first? Evaluate.
What happens differently here than in our last example?
What should our next step be?
Evaluate to find the final answer.

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What do you noce about the two answers?
What was different between the two expressions?
What conclusions can you draw about evaluang expressions with parentheses and
exponents?
2 x (3 + 42) = 38 2 x (3 + 4)2 = 98

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Exercises
8. 7 + (12 – 32) 9. 7 + (12 – 3)2

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Closing
Please take out your exit ticket for Lesson 6, close your
binder, and complete the exit ticket. This will be collected.

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